A Comparative Study of High-Order, Quasi-Conservative, Semi-Lagrangian, and Eulerian Advection Schemes

Ching-Yuang Huang Department of Atmospheric Sciences, National Central University, Chung-Li, Taiwan

Search for other papers by Ching-Yuang Huang in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

Quasi-conservative high-order semi-Lagrangian advection schemes are compared with several positive-definite Eulerian schemes in flux form, including Bott’s scheme. In this study, the conventional equipartition method is modified as a posterior iterative mass correction algorithm to restore conservation for semi-Lagrangian transport. The performance comparisons between semi-Lagrangian and Eulerian schemes are evidence that the fifth-order (seventh-order) Bott’s area-preserving algorithm without flux limitation is practically equivalent to the quintic (seventh-order) semi-Lagrangian scheme in the rotational flow, with the maximum directional Courant number smaller than 0.5. For positive-definite advection, Bott’s algorithm with the flux limitation obtains slightly better (worse) amplitude preservation in the rotational flow tests compared to semi-Lagrangian schemes of same order with (without) the mass correction algorithm. In the nonlinear deformational flow where the maximum directional Courant number is greater than 0.5, the former is slightly unstable and only the short-term results are acceptable, but the latter at the long-term remains stable, conservative, and reasonable.

Monotonic tests of semi-Lagrangian schemes were also conducted. It was found that quasi-monotone schemes based on a posterior monotonicity constraint are not influenced by the mass correction algorithm. Both monotonicity and mass conservation can be achieved simultaneously for semi-Lagrangian transport by the post adjustments. However, the monotonicity constraint itself cannot fully suppress the numerical dispersion in the strong nonlinear deformational flow where the mass correction procedure appears to be significantly important for strict mass conservation and a reduction in phase errors.

Corresponding author address: Ching-Yuang Huang, Dept. of Atmospheric Sciences, National Central University, 32054 Chung-Li, Taiwan.

In%“T233217@twncu865.ncu,edu.tw

Abstract

Quasi-conservative high-order semi-Lagrangian advection schemes are compared with several positive-definite Eulerian schemes in flux form, including Bott’s scheme. In this study, the conventional equipartition method is modified as a posterior iterative mass correction algorithm to restore conservation for semi-Lagrangian transport. The performance comparisons between semi-Lagrangian and Eulerian schemes are evidence that the fifth-order (seventh-order) Bott’s area-preserving algorithm without flux limitation is practically equivalent to the quintic (seventh-order) semi-Lagrangian scheme in the rotational flow, with the maximum directional Courant number smaller than 0.5. For positive-definite advection, Bott’s algorithm with the flux limitation obtains slightly better (worse) amplitude preservation in the rotational flow tests compared to semi-Lagrangian schemes of same order with (without) the mass correction algorithm. In the nonlinear deformational flow where the maximum directional Courant number is greater than 0.5, the former is slightly unstable and only the short-term results are acceptable, but the latter at the long-term remains stable, conservative, and reasonable.

Monotonic tests of semi-Lagrangian schemes were also conducted. It was found that quasi-monotone schemes based on a posterior monotonicity constraint are not influenced by the mass correction algorithm. Both monotonicity and mass conservation can be achieved simultaneously for semi-Lagrangian transport by the post adjustments. However, the monotonicity constraint itself cannot fully suppress the numerical dispersion in the strong nonlinear deformational flow where the mass correction procedure appears to be significantly important for strict mass conservation and a reduction in phase errors.

Corresponding author address: Ching-Yuang Huang, Dept. of Atmospheric Sciences, National Central University, 32054 Chung-Li, Taiwan.

In%“T233217@twncu865.ncu,edu.tw

Save
  • Bermejo, R., 1990: On the equivalence of semi-Lagrangian schemes and particle-in-cell finite element method. Mon. Wea. Rev.,118, 979–987.

  • ——, and A. Staniforth, 1992: The conversion of semi-Lagrangian advection schemes to quasi-monotone schemes. Mon. Wea. Rev.,120, 2622–2632.

  • Bott, A., 1989a: A positive definite advection scheme obtained by nonlinear renormalization of the advection flux. Mon. Wea. Rev.,117, 1006–1015.

  • ——, 1989b: Reply. Mon. Wea. Rev.,117, 2633–2636.

  • ——, 1992: Monotone flux limitation in the area-preserving flux-form advection algorithm. Mon. Wea. Rev.,120, 2592–2602.

  • Carpenter, R. L., Jr., K. K. Drogemeier, P. R. Woodward, and C. E. Hane, 1990: Application of the piecewise parabolic method to meteorological modeling. Mon. Wea. Rev.,118, 586–612.

  • Chlond, A., 1994: Locally modified version of Bott’s advection scheme. Mon. Wea. Rev.,122, 111–125.

  • Colella, P., and P. R. Woodward, 1984: The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys.,54, 174–201.

  • Crowley, W. P., 1968: Numerical advection experiments. Mon. Wea. Rev.,96, 1–11.

  • Easter, R. C., 1993: Two modified versions of Bott’s positive-definite numerical advection scheme. Mon. Wea. Rev.,121, 297–304.

  • Emde, V. D., 1992: Solving conservation laws with parabolic and cubic splines. Mon. Wea. Rev.,120, 482–492.

  • Gravel, S., and A. Staniforth, 1992: Variable resolution and robustness. Mon. Wea. Rev.,120, 2633–2640.

  • ——, and ——, 1994: A mass-conserving semi-Lagrangian scheme for the shallow-water equations. Mon. Wea. Rev.,122, 243–248.

  • Huang, C. Y., 1993: A study of high-order advectionschemes in variable resolution. Terr. Atmos. Ocean,4, 421–440.

  • ——, 1994: Semi-Lagrangian advection schemes and Eulerian WKL algorithms. Mon. Wea. Rev.,122, 1647–1658.

  • ——, and S. Raman, 1991: A comparative study of numerical advection schemes featuring a one-step modified WKL algorithm. Mon. Wea. Rev.,119, 2900–2918.

  • Krishnamurti, T. N., 1962: Numerical integration of primitive equations by a quasi-Lagrangian advection scheme. J. Appl. Meteor.,1, 508–521.

  • Lin, S.-J., W. C. Chao, Y. C. Sud, and G. K. Waker, 1994: A class of the van Leer-type transport schemes and its application to the moisture transport in a general circulation model. Mon. Wea. Rev.,122, 1575–1593.

  • Long, P. E., Jr., and D. W. Pepper, 1981: An examination of some simple numerical schemes for calculating scalar advection. J. Appl. Meteor.,20, 146–156.

  • Makar, P. A., and S. R. Karpik, 1996: Basis-spline interpolation on the sphere: Applications to semi-Lagrangian advection. Mon. Wea. Rev.,124, 182–199.

  • McGregor, J. L., 1993: Economical determination of departure points for semi-Lagrangian models. Mon. Wea. Rev.,121, 221–230.

  • Navon, I. M., 1981: Implementation of a posteriori methods for enforcing conservation of potential enstrophy and mass in discretized shallow-water equations. Mon. Wea. Rev.,109, 946–958.

  • Priestley, A., 1993: A quasi-conservative version of the semi-Lagrangian advection scheme. Mon. Wea. Rev.,121, 621–629.

  • Pudykiewicz, J., and A. Staniforth, 1984: Some properties and comparative performance of the semi-Lagrangian method of Robert in the solution of the advection-diffusion equation. Atmos.–Ocean,22, 283–308.

  • Purnell, D. K., 1976: Solution of the advective equation by upstream interpolation with a cubic spline. Mon. Wea. Rev.,104, 42–48.

  • Rasch, P. J., and D. L. Williamson, 1990: On shape-preserving interpolation and semi-Lagrangian transport. SIAM J. Sci. Stat. Comput.,11, 656–687.

  • Smolarkiewicz, P. K., 1982: The multidimensional Crowley advection scheme. Mon. Wea. Rev.,110, 1968–1983.

  • ——, 1983: A simple positive definite advection scheme with small implicit diffusion. Mon. Wea. Rev.,111, 479–486.

  • ——, 1984: A full multidimensional positive definite advection algorithm with small implicit diffusion. J. Comput. Phys.,54, 325–362.

  • ——, 1989: Comment on “A positive definite advection scheme obtained by nonlinear renormalization of the advective fluxes.” Mon. Wea. Rev.,117, 2626–2636.

  • ——, and T. L. Clark, 1986: A full multidimensional positive definite advection algorithm: Further development and applications. J. Comput. Phys.,67, 396–438.

  • ——, and P. J. Rasch, 1991: Monotone advection on the sphere: An Eulerian versus semi-Lagrangian approach. J. Atmos. Sci.,48, 793–810.

  • Staniforth, A., and J. Côté, 1991: Semi-Lagrangian integration schemes for atmospheric models—A review. Mon.Wea. Rev.,119, 2206–2223.

  • ——, ——, and J. Pudykiewicz, 1987: Comments on “Smolarkiewicz’s deformation flow.” Mon. Wea. Rev.,115, 894–900.

  • Tremback, C. J., J. Powell, W. R. Cotton, and R. A. Pielke, 1987: The forward-in-time upstream advection scheme: Extension to higher orders. Mon. Wea. Rev.,115, 540–555.

  • van Leer, B., 1977: Toward the ultimate conservative difference scheme. IV: A new approach to numerical convection. J. Comput. Phys.,23, 276–299.

  • ——, 1979: Toward the ultimate conservative difference scheme. V: A second order sequel to Godunov’s method. J. Comput. Phys.,32, 101–136.

  • Williamson, D. L., and P. J. Rasch, 1989: Two-dimensional semi-Lagrangian transport with shape preserving interpolation. Mon. Wea. Rev.,117, 102–129.

  • ——, and ——, 1994: Water vapor transport in the NCAR CCM2. Tellus,46A, 34–51.

  • Zalesak, S. T., 1979: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys.,31, 335–362.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 176 25 1
PDF Downloads 33 11 0